Finite Group Databasebeta

← #46845841

Subgroups — #46845841

Order 34688.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576187C212yesmaximal, complemented
2#127×D271C211yesmaximal, characteristic, complemented
3C4×#16864655C211yesmaximal, characteristic, complemented
4#31774097C211yesmaximal, characteristic, complemented
5#31774095C212yesmaximal, complemented
6#31774093C211yesmaximal, characteristic, complemented
7#31774091C212yesmaximal, complemented
8#31774089C211yesmaximal, characteristic, complemented
9#31773919C211yesmaximal, characteristic, complemented
10#31773900C212yesmaximal, complemented
11#460×C271C211yesmaximal, characteristic, complemented
12C2×#16864655C2211yescharacteristic
13C2×D4×D271—21no—
14C2×D4×D271—21no—
15C2×D4×D271C2212yes—
16#22576187—28no—
17C2×#16864641C2212yescomplemented
18C2×D2168C2212yescomplemented
19C2×#16864639C2212yescomplemented
20C2×#16864637C2212yes—
21#22576157C2211yescharacteristic
22#22576156—21no—
23#22576156—21no—
24#22576156C2212yes—
25Q8×Dic271C2211yescharacteristic, complemented
26C4×D1084—21no—
27C4×D1084C2211yescharacteristic, complemented
28#22576122—21no—
29#22576122C2211yescharacteristic, complemented
30#22576118—21no—
31#22576118C2211yescharacteristic
32C42×D271—21no—
33C42×D271C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic, complemented
35Dic271⋊C8C2212yescomplemented
36#22576082C2212yescomplemented
37#22576081C2212yescomplemented
38#22576080C2212yescomplemented
39C2×SD16×C271C2212yescomplemented
40#127×C271C2211yescharacteristic
41C4×Q8×C271C2211yescharacteristic, complemented
42C4⋊C8×C271C2211yescharacteristic, complemented
43#102×C271C2212yescomplemented
44C23×D271—22no—
45#16864655—22no—
46#16864655D414yescomplemented
47C2×D1084—21no—
48C2×D1084—21no—
49C2×D1084—21no—
50C2×D1084C2312yes—
51C2×D1084C2311yescharacteristic
52C2×C22⋊D271—22no—
53C2×C22⋊D271—22no—
54C2×C22⋊D271—22no—
55D4×D271—24no—
56D4×D271—24no—
57D4×D271—44no—
58D4×D271—44no—
59C2×C4×D271—21no—
60C2×C4×D271—21no—
61C2×C4×D271—21no—
62C2×C4×D271D411yescharacteristic
63C2×C4×D271D411yescharacteristic
64C2×C4×D271C2311yescharacteristic
65#16864641—24no—
66D2168—24no—
67#16864639—24no—
68#16864637—24no—
69C2×C271⋊C8C2312yes—
70C4⋊Dic271—21no—
71C4⋊Dic271C2311yescharacteristic
72(C2×D271)⋊C4—21no—
73(C2×D271)⋊C4—21no—
74(C2×D271)⋊C4—21no—
75C4×Dic271—21no—
76C4×Dic271—21no—
77C4×Dic271D411yescharacteristic
78C4×Dic271D411yescharacteristic
79C4×Dic271C2311yescharacteristic
80C2×Q8×C271C2311yescharacteristic
81C2×D4×C271—21no—
82C2×D4×C271—21no—
83C2×D4×C271C2312yes—
84SD16×C271—24no—
85C2×C8×C271C2312yes—
86C4⋊C4×C271—21no—
87C4⋊C4×C271C2311yescharacteristic
88C42×C271—21no—
89C42×C271C2311yescharacteristic
90C22×D271—21no—
91C22×D271—22no—
92C22×D271—44no—
93C22×D271—44no—
94C22×D271—44no—
95C22×D271C2×D411yescharacteristic
96D1084—24no—
97D1084—22no—
98D1084—21no—
99D1084—22no—
100D1084—42no—
101D1084—42no—
102D1084C2×D412yescomplemented
103C22⋊D271—44no—
104C22⋊D271—44no—
105C22⋊D271—44no—
106C2×Dic271—21no—
107C2×Dic271C2×D411yescharacteristic
108C2×Dic271C2×D411yescharacteristic
109C2×Dic271C2×D411yescharacteristic
110C4×D271—22no—
111C4×D271—24no—
112C4×D271—22no—
113C4×D271—42no—
114C4×D271—42no—
115C4×D271C2×D412yes—
116C4×D271#4112yes—
117C271⋊C8—22no—
118C23×C271—22no—
119Q8×C271—21no—
120Q8×C271C2×D412yescomplemented
121D4×C271—22no—
122D4×C271—22no—
123D4×C271—42no—
124D4×C271—42no—
125C2×C4×C271—21no—
126C2×C4×C271—21no—
127C2×C4×C271C2×D411yescharacteristic
128C2×C4×C271C2×D411yescharacteristic
129C2×C4×C271C2411yescharacteristic
130C8×C271—22no—
131C22×C271—42no—
132C22×C271—42no—
133C22×C271—42no—
134C22×C271C22×D411yescharacteristic
135C2×D271—21no—
136C2×D271—21no—
137C2×D271—22no—
138C2×D271—41no—
139C2×D271—41no—
140C2×D271—42no—
141C2×D271—42no—
142C2×D271—42no—
143C2×D271—88no—
144C2×D271#12112yes—
145C4×C271—22no—
146C4×C271—21no—
147C4×C271—41no—
148C4×C271AGL(1,ℤ/8)12yescomplemented
149C4×C271C22×D411yescharacteristic
150C4×C271C2×#4111yescharacteristic
151Dic271—21no—
152Dic271—41no—
153Dic271#12112yes—
154Dic271AGL(1,ℤ/8)12yescomplemented
155C2×C271—82no—
156C2×C271C2×#12111yescharacteristic
157C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
159D271—22no—
160D271—42no—
161D271—82no—
162C271#303811yescharacteristic, complemented, Sylow
163#3038—2711nomaximal, Sylow
164C2×AGL(1,ℤ/8)—2712no—
165C2×#127—2711no—
166C4×#41—2711no—
167#460—2712no—
168#460—2711no—
169#460D27111yescharacteristic, complemented
170#459—2711no—
171#459—2711no—
172#459—2712no—
173#423—2711no—
174#418—2712no—
175C2×#41—2711no—
176C22×D4—2712no—
177C22×D4—5421no—
178C22×D4—5421no—
179AGL(1,ℤ/8)—5428no—
180C2×SD16—2712no—
181C2×SD16C2×D27112yescomplemented
182C2×D8—2712no—
183C2×D8—2712no—
184C2×M16—2712no—
185#127—2712no—
186#127—2711no—
187#127—5421no—
188#127—5421no—
189#127C2×D27111yescharacteristic
190C4×Q8—2711no—
191C4×Q8C2×D27111yescharacteristic, complemented
192C4×D4—2711no—
193C4×D4—2711no—
194C4×D4—5421no—
195C4×D4—5421no—
196#117—2711no—
197#117—5421no—
198C2×C42—2711no—
199C2×C42—5421no—
200C4⋊C8—2712no—
201C4⋊C8—2711no—
202C4⋊C8C2×D27111yescharacteristic, complemented
203#102—2712no—
204#102—2712no—
205#102—2712no—
206#102C2×D27112yescomplemented
207C24—5422no—
208#41—2714no—
209#41—5422no—
210C2×Q8C22×D27111yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—2711no—
214C2×D4—2712no—
215C2×D4—5421no—
216C2×D4—5424no—
217C2×D4—5424no—
218C2×D4—5422no—
219C2×D4—5422no—
220C2×D4—5422no—
221C2×D4—5421no—
222C2×D4—5421no—
223C2×D4—10844no—
224C2×D4—10844no—
225C2×D4C22×D27112yes—
226C22×C4—2711no—
227C22×C4—2711no—
228C22×C4—2711no—
229C22×C4—5421no—
230C22×C4—5421no—
231C22×C4—5421no—
232SD16—24no—
233SD16—5424no—
234D8—5424no—
235D8—5424no—
236M16—5424no—
237C2×C8—2712no—
238C2×C8C22×D27112yes—
239C4⋊C4—21no—
240C4⋊C4—2711no—
241C4⋊C4—5421no—
242C4⋊C4C22×D27111yescharacteristic
243C22⋊C4—5421no—
244C22⋊C4—5421no—
245C22⋊C4—5421no—
246C42—21no—
247C42—2711no—
248C42—2711no—
249C42—2711no—
250C42—5421no—
251C42—5421no—
252C42C22×D27111yescharacteristic
253C23—22no—
254C23—2711no—
255C23—5421no—
256C23—5422no—
257C23—10844no—
258C23—10844no—
259C23—10844no—
260Q8—21no—
261Q8D4×D27112yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—2712no—
267D4—5424no—
268D4—5422no—
269D4—5421no—
270D4—5422no—
271D4—10842no—
272D4—10844no—
273D4—10844no—
274D4—10844no—
275D4—10842no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—2711no—
279C2×C4—2711no—
280C2×C4—2712no—
281C2×C4—2712no—
282C2×C4—2711no—
283C2×C4—5422no—
284C2×C4—5424no—
285C2×C4—5422no—
286C2×C4—5421no—
287C2×C4—10842no—
288C2×C4—10842no—
289C2×C4D4×D27111yescharacteristic
290C2×C4D4×D27111yescharacteristic
291C2×C4C23×D27111yescharacteristic
292C8—22no—
293C8—5422no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—2712no—
298C22—5421no—
299C22—5421no—
300C22—5422no—
301C22—10841no—
302C22—10841no—
303C22—10842no—
304C22—10842no—
305C22—10842no—
306C22—21688no—
307C22C2×D4×D27111yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—2712no—
312C4—2712no—
313C4—5421no—
314C4—10841no—
315C4#2257618712yescomplemented
316C4C2×D4×D27111yescharacteristic
317C4#41×D27111yescharacteristic
318C2—82no—
319C2—5422no—
320C2—10842no—
321C2—21682no—
322C2#121×D27111yescharacteristic
323C2AGL(1,ℤ/8)×D27111yescharacteristic
324C2C2×#2257618711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#460×C271C2
socleC22×C271C22×D4
hypercenter#460D271
hypocenterC271#3038
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. #127×D271C2semidirect
3. C4×#16864655C2semidirect
4. #31774097C2semidirect
5. #31774095C2semidirect
6. #31774093C2semidirect
7. #31774091C2semidirect
8. #31774089C2semidirect
9. #31773919C2semidirect
10. #31773900C2semidirect
11. #460×C271C2semidirect
17. C2×#16864641C22semidirect
18. C2×D2168C22semidirect
19. C2×#16864639C22semidirect
25. Q8×Dic271C22semidirect
27. C4×D1084C22semidirect
29. #22576122C22semidirect
34. C4⋊(C271⋊C8)C22semidirect
35. Dic271⋊C8C22semidirect
36. #22576082C22semidirect
37. #22576081C22semidirect
38. #22576080C22semidirect
39. C2×SD16×C271C22semidirect
41. C4×Q8×C271C22semidirect
42. C4⋊C8×C271C22semidirect
43. #102×C271C22semidirect
46. #16864655D4semidirect
102. D1084C2×D4semidirect
120. Q8×C271C2×D4semidirect
148. C4×C271AGL(1,ℤ/8)semidirect
154. Dic271AGL(1,ℤ/8)semidirect
162. C271#3038semidirect
169. #460D271semidirect
181. C2×SD16C2×D271semidirect
191. C4×Q8C2×D271semidirect
202. C4⋊C8C2×D271semidirect
206. #102C2×D271semidirect
261. Q8D4×D271semidirect
315. C4#22576187semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#225761872
#127×D2712
C4×#168646552
#317740972
#317740952
#317740932
#317740912
#317740892
#317739192
#317739002
#460×C2712
#3038271